1. (a + b) 0 =1 (a + b) 1 = (a + b) 2 = (a + b) 3 = 1a 1 + 1b 1 1a 2 + 2ab + 1b 2 1a 3 + 3a 2 b + 3ab 2 + 1b 3 Binomial Expansion... What do we notice????

2. Pascal’s Triangle

3. Example Expand (a + b) 6 Solution: (a + b) 6 = a 6 + 6a 5 b + 15a 4 b 2 + 20a 3 b 3 + 15a 2 b 4 + 6ab 5 + b 6

4. Example Find the first four terms in the expansion of (x – 2y) 7. Solution (a + b) 7 = a 7 + 7a 6 b + 21a 5 b 2 + 35a 4 b 3 + ….. Substitute: x = a, 2y = b (x + 2y) 7 = x 7 + 7x 6 (2y) + 21x 5 (2y) 2 + 35x 4 (2y) 3 +... Simplify (x + 2y) 7 = x 7 + 14x 6 y + 84x 5 y 2 + 280x 4 y 3

5. Example a 12 + 12a 11 b + 66a 10 b 2 + 220a 9 b 3

7. Factorial Notation r! = r(r-1)(r-2)(r-3)…..3*2*1 9! = 9*8*7*6*5*4*3*2*1 = 362880 12! = 12*11*10*9*8*7*6*5*4*3*2*1 = 479001600 What happens when we divide factorials? Calculators can help! MATH -> PRB -> #4

8. Therefore, you can write the fourth term of the expansion of (a + b) 12 as follows: Please write a formula for the k term of the expansion of (a + b) n.

9. Example Find and simplify the seventh term in the expansion of (2x-y) 10. 210(2x) 4 (-y) 6 = 210 * 16x 4 * y 6 = 3360x 4 y 6